Right-reduction enables the computation of pure-graded PH cycle bases and PrcH cocycle bases via the following result of [1]}, [2]}. For convenience, given a right-reduction \(Y\) , write \({E}_{Y}\)  for the invertible upper triangular matrix such that

Colj( EY )
=
{ll
Coli(R)  j = lowR(i)  for some  i
